## Welcome

Prof. (assoc.) dr. habil. László Szilárd Csaba

Technical University of Cluj-Napoca

Memorandumului nr. 28 ,

400114 Cluj-Napoca,  Romania

e-mail: laszlosziszi@yahoo.com

Publications

Szilárd László
ISBN  978-3659497636

The aim of this work is to present several new results concerning monotone operator theory, applications  to injectivity theorems, convex functions and variational inequalities, and some applications of the duality to the theory of maximal monotone operators.

We begin with the study under which circumstances the local (generalized) monotonicity property of an operator is equivalent to its global counterpart.

We introduce two new concepts, the θ-monotonicity of operators and the θ-convexity of real valued functions, that contain in particular several monotonicity/convexity notions from the literature.

We dedicate a whole chapter to the study of variational inequalities of Stampacchia and Minty type and we establish several coincidence, respectively fixed point results.

The last part this work is devoted to the study of famous sum problems involving two maximal monotone operators. Based on stable strong duality and introducing some generalized infimal convolution formulas, we establish most of the important results obtained so far in the literature related to this problem.

For students

László Szilárd
Csaba

Elements of Linear Algebra
ISBN 978-973-662-935-8

The aim of this book is to give an introduction in linear algebra, and at the same time to provide some applications that might be useful both in practice and theory. Hopefully this book will be a real help for graduate level students to understand the basics of this beautiful mathematical field called linear algebra. Our scope is twofold: one hand we give a theoretical introduction to this field, which is more than exhaustive for the need and understanding capability of graduate students, on the other hand we present fully solved examples and problems, that might be helpful in preparing to exams and also show the techniques used in the art of problem solving on this field. At the end of every chapter, this work contains several proposed problems that can be solved using the theory and solved examples presented previously. The book is structured in seven chapters, we begin with the study of matrices and determinants and we end with the study of quadratic forms with some useful applica- tions in analytical geometry.  Our hope is that our students, and not just them, will enjoy studying this book and they will find very helpful in preparing for exams.

Editura U.T.PRESS, Str.Observatorului nr. 34 C.P.42, O.P. 2,  400775 Cluj-Napoca
Tel.:0264-401.999 / Fax: 0264 - 430.408,  e-mail: utpress@biblio.utcluj.ro www.utcluj.ro/editura
Director: Prof.dr.ing. Daniela Manea
Consilier editorial: Ing. Călin D. Câmpean

Journal papers

1. G. Kassay, C. Pintea, Szilárd László: Monotone operators and closed countable sets. Optimization 2011; 60(8-9):1059-1069.
ABSTRACT: In this article we prove that the local increasing monotonicity of an operator on the complement of a certain type of closed set implies its global increasing monotonicity. As applications we obtain some global injectivity/convexity results based on local injectivity/ convexity properties and some extra analytic requirements.

2. Szilárd László: Some Existence Results of Solutions for General Variational Inequalities. Journal of Optimization Theory and Applications 2011; 150(1):425-443.
ABSTRACT: In this paper, we introduce a new class of operators. We present some fundamental properties of the operators belonging to this class and, as applications, we establish some existence results of the solutions for several general variational inequalities involving elements belonging to this class.

3. Szilárd László: Generalized Monotone Operators, Generalized Convex Functions and Closed Countable Sets. Journal of Convex Analysis 2011; 18:1075-1091.
ABSTRACT: In this paper we deal with operators which are monotone in several generalized sense. We show that if such property holds locally on the complement of a certain type of closed set, then the same property holds globally on the whole domain under some mild conditions. Our results extend similar statements already established for the classical Minty-Browder monotonicity. As applications we obtain some global generalized convexity results based on local generalized convexity property and some extra analytical requirements.

4. Szilárd László: Theta-monotone operators and theta-convex functions. Taiwanese Journal of Mathematics 2012; 16:733-759.
ABSTRACT: In this paper we introduce a new monotonicity concept for multivalued operators, respectively, a new convexity concept for real valued functions, which generalize several monotonicity, respectively, convexity notions already known in literature. We present some fundamental properties of the operators having this monotonicity property. We show that if such a monotonicity property holds locally then the same property holds globally on the whole domain of the operator. We also show that these two new concepts are closely related. As an immediate application we furnish some surjectivity results in finite dimensional spaces.

5. Szilárd László, B. Burjan-Mosoni: About the Maximal Monotonicity of the Generalized Sum of Two Maximal Monotone Operators. Set-Valued and Variational Analysis 2012; 20(3):355-368.
ABSTRACT: We give several regularity conditions, both closedness and interior point type, that ensure the maximal monotonicity of the generalized sum of two strongly representable monotone operators, and we extend some recent results concerning on the same problem.

6. Radu Ioan Boţ, Szilárd László: On the generalized parallel sum of two maximal monotone operators of Gossez type (D). Journal of Mathematical Analysis and Applications 2012; 391(1):82-98.
ABSTRACT: The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This result will follow as a particular instance of a more general one concerning the maximal monotonicity of Gossez type (D) of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals.

7. Szilárd László: Existence of solutions of inverted variational inequalities. Carpathian Journal of Mathematics 2012; 28(2):271-278.
ABSTRACT: In this paper we introduce two new generalized variational inequalities and we give some existence results of the solutions for these variational inequalities involving operators belonging to a recently introduced class of operators. We show by examples, that our results fail outside of this class. Further, we establish a result that may be viewed as a generalization of Minty’s theorem, that is, we show that under some circumstances the set of solutions of these variational inequalities coincide. We also show, the condition that the operators, involved in these variational inequalities, belong to the above mentioned class, is essential in obtaining this result. As application, we show that Brouwer’s fixed point theorem is an easy consequence of our results.

8. G. Kassay, C. Pintea, Szilárd László: Monotone operators and first category sets. Positivity 2012; 16(3):565-577.
ABSTRACT: In this paper we show that the local monotonicity in the sense of Minty and Browder on some residual sets assure the global monotonicity and, according to an earlier result, the convexity of the inverse images. We pay some special attention to the residual sets arising as complements of some special first Baire category sets, namely the σ-affine sets, the σ-compact sets and the σ-algebraic varieties.We achieve this goal gradually by showing, at first, that the continuous real valued functions of one real variable, which are locally nondecreasing on sets whose complements have no nonempty perfect subsets, are globally nondecreasing. The convexity of the inverse images combined with their discreteness, in the case of local injective operators, ensure the global injectivity. Note that the global monotonicity and the local injectivity of regular enough operators is guaranteed by the positive definiteness of the symmetric part of their Gâteaux differentials on the involved residual sets. We close this work with a short subsection on the global convexity which is obtained out of its local counterpart on some residual sets.

9. Szilárd László: Multivalued variational inequalities and coincidence point results. Journal of Mathematical Analysis and Applications 2013; 404(1):105-114.
ABSTRACT: In this paper, we establish some existence results of the solutions for several multivalued variational inequalities involving elements belonging to a class of operators that was recently introduced in literature. As applications we obtain some new coincidence point results in Hilbert spaces.

10. Szilárd László: A bivariate infimal convolution formula and the maximal monotonicity of the parallel sum. Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity 2013; 11:3-29.
ABSTRACT: By making use of some well known infimal convolution formulas, we give several regularity conditions, both closedness and interior point type, that ensure the maximal monotonicity of the parallel sum of two maximal monotone operators of Gossez type (D) in arbitrary Banach spaces. We also obtain similar results for the parallel sum of the Gossez monotone closure of two maximal monotone operators.

11. A. Amini-Harandi, Szilárd László: A coincidence point result via variational inequalities. Fixed Point Theory 2014; 15(1):87-98.
ABSTRACT: In this paper, by making use of a new class of operators, we establish some existence results of the solution for an extended general variational inequality already considered in the literature. As application we obtain a new coincidence point theorem in a Hilbert space setting.

12. Szilárd László: On the strong representability of the generalized parallel sum. The Bulletin of the Malaysian Mathematical Society Series 2, 2014; 37(4):1029-1046
ABSTRACT: We give several regularity conditions, both closedness and interior point type, that ensure the maximal monotonicity of the generalized parallel sum of two maximal monotone operatorsof Gossez type (D), and we extend some recent results concerning on the same problem.

13. A. Amini-Harandi, Szilárd László: Solution existence of general variational inequalities and coincidence points. Carpathian Journal of Mathematics 2014; 30(1):9-17.
ABSTRACT: In this paper, by using a simple technique, we obtain several existence results of the solutions for general variational inequalities of Stampacchia type. We also show, that the existence of a coincidence point of two mappings is equivalent to the existence of the solution of a particular general variational inequality of Stampacchia type. As applications several coincidence and fixed point results are obtained.

14. A. Amini-Harandi, Szilárd László: Applications of general variational inequalities to coincidence point results. Publicationes Mathematicae Debrecen 2014; 85(1-2):47-58.
ABSTRACT: In this paper we obtain some existence result of solution for general variational inequalities. As applications several coincidence and fixed point results are provided.

15. Szilárd László, A. Viorel: Generalized monotone operators on dense sets. Numerical Functional Analysis and  Optimization 2015; 36(7): 901-929.
ABSTRACT: In the present work we show that under mild assumptions the local generalized monotonicity of a set-valued operator on a special type of dense set, that we call self segment-dense, ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the claim holds in the case of set-valued functions of one real variable which are locally generalized monotone on a dense subset of their domain. Then, we extend our proof to the case of set-valued operators on arbitrary Banach spaces, imposing only a convexity requirement on the domain of the operator. We apply these results in order to obtain the global generalized convexity of a functional out of the local property on a self segment-dense set. Further we give some useful application of the self segment-dense sets in the framework of equilibrium problems.

16.  Szilárd László, Adrian Viorel: Densely defined equilibrium problemsJournal of Optimization Theory and Applications 2015; 166(1): 52-75.
ABSTRACT:
In the present work we deal with set-valued equilibrium problems for which we provide sufficient conditions for the existence of a solution. The conditions that we consider are imposed not on the whole domain, but rather on a self segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu-Gale-Nikaido type theorem, with a considerable weakened Walras law in its hypothesis. Further, we consider a non-cooperative n-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.

17. Radu Ioan Boț, Ernő Robert Csetnek, Szilárd László: An inertial forward-backward algorithm for minimizing the sum of two non-convex functions, Euro Journal on Computational Optimization, 2016;  4(1), 3-25..
ABSTRACT: We propose a forward-backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. The sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-Lojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions. We illustrate the theoretical results by considering two numerical experiments: the first one concerns the ability of recovering the local optimal solutions of nonconvex optimization problems, while the second one refers to the restoration of a noisy blurred image.

18. Szilárd László: On injectivity of a class of monotone operators with some univalency consequences. Mediterranean Journal of Mathematics 2016; 13(2):       729-744.
ABSTRACT: In this paper we provide sufficient conditions that ensure the global injectivity of an operator. Further, some new analytical conditions that assure the injectivity/univalence of a complex function of one complex variable are obtained. We also show that some classical results, such as Alexander-Noshiro-Warschawski and Wolff theorem or Mocanu theorem, are easy consequences of our results.

19.   Szilárd László: Vector equilibrium problems on dense sets, Journal of Optimization Theory and Applications 2016; 170(2): 437-457.
ABSTRACT: In this paper we provide sufficient conditions that ensure the existence of the solution of some vector equilibrium problems in Hausdorff topological vector spaces ordered by a cone. The conditions that we consider are imposed not on the whole domain of the operators involved, but rather on a self segment-dense subset of it, a special type of dense subset. We apply the results obtained to vector optimization and vector variational inequalities.

20. Szilárd László: Minimax Results on Dense Sets and Dense Families of Functionals, Siam Journal on Optimization 2017, 27(2): 661-685.
ABSTRACT: In this paper we deal with minimax results on dense sets. We study at first under which conditions the infimum of a function over a dense subset of its domain, coincides with the global infimum of that function. Then, we apply our results in order to obtain several minimax results on dense sets. Finally, we obtain the denseness of some parameterized families of functionals in the Banach  space of bounded functions and we provide an alternative proof of the famous reflexivity result of James.

21. R.I. Boț, E.R. Csetnek, Szilárd László: A second order dynamical system and a monotone inclusion problem, Analysis and Applications 2018, 16(5):601-622 pdf.
ABSTRACT: In this paper we consider a second order dynamical system of the from $\ddot{x}(t)+\g(t)\dot{x}(t)+x(t)-J_{\l(t) A}(x(t)-\l(t) D(x(t))-\l(t)\b(t)B(x(t)))=0$, where $A:{\mathcal H}\toto{\mathcal H}$ is a maximal monotone operator, $J_{\l(t) A}:{\mathcal H}\To{\mathcal H}$ is the resolvent operator of $\l(t)A$, $D,B: {\mathcal H}\rightarrow{\mathcal H}$ are cocoercive operators defined on a real Hilbert space ${\mathcal H}$, $\lambda,\beta:[0,+\infty)\rightarrow [0,+\infty)$ are relaxation functions  and $\gamma:[0,+\infty)\rightarrow [0,+\infty)$ a damping function, all depending on time. We show the existence and uniqueness of
strong global solutions in the framework of the Cauchy-Lipschitz-Picard Theorem and prove ergodic asymptotic convergence for the generated trajectories to a zero of the operator $A+D+{N}_C,$ where $C=\zer(B)$ and $N_C$ is the normal cone operator, by using Lyapunov analysis combined with the celebrated Opial Lemma in its ergodic continuous version. Furthermore, we show the strong convergence of trajectories to the unique zero of $A+D+{N}_C$ in case $A$ is a strongly monotone operator. The framework allows to address as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one and allows us to recover and improve several results from the literature.

22. R.I. Boț, E.R. Csetnek, Szilárd László:  Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems
, pdf, (accepted) Journal of Evolution Equations, doi:10.1007/s00028-018-0441-7
ABSTRACT: We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection  with the minimization of the sum of a nonsmooth convex  and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Lojasiewicz exponent.

23. Szilárd László: A primal-dual approach of weak vector equilibrium problems, Open Mathematics 2018, 16:276-288,  pdf.
ABSTRACT: In this paper we provide some new sufficient conditions that ensure the existence of the solution of a weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone.  Further, we introduce a dual problem and we provide conditions that assure the solution set of the original problem and its dual coincide. We show that many known problems from the literature can be treated in our primal-dual model. We provide several coercivity conditions in order to obtain solution existence of the primal-dual problems without compactness assumption. We pay a special attention to the case when the base space is a reflexive Banach space.  We apply the results obtained to perturbed vector equilibrium problems.

24. Szilárd László: On .perturbed weak vector  equilibrium problems under new semi-continuities, pdf, https://arxiv.org/abs/1708.06651
ABSTRACT: In this paper we introduce a new semicontinuity notion, which is weaker than upper semicontinuity, and assures the closedness of the sets $G(y)=\{x\in K: f(x,y)\not\in -\inte C\}.$ Furhter, this semicontinuity is also closed under addition. These two properties make our new semicontinuity applicable in situations where other semicontinuities, like quasi upper semicontinuity or order upper semicontinuity, fail. The above emphasized properties are some key tools in order to provide new sufficient conditions that ensure the existence of the solution of a perturbed weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone. Further, we introduce a dual problem and we provide conditions that assure that every solution of the dual problem is also a solution of the perturbed weak vector equilibrium problem. We apply the results obtained to Ekeland vector variational principles.

25. R.I. Boț, E.R. Csetnek,  Szilárd László: A second order dynamical  system with vanishing coefficients associated to a nonconvex minimization, pdf
ABSTRACT: In this paper we study a second order dynamical system with variable coefficients in connection  with the minimization of a smooth nonconvex function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Lojasiewicz exponent.

26. R.I. Boț, E.R. Csetnek,  Szilárd László:  A second order dynamical approach with variable damping to nonconvex smooth minimization, Applicable Analysis, accepted 2018pdf
ABSTRACT: We investigate a second order dynamical system with variable damping in connection with the minimization of a nonconvex differentiable function. The dynamical system is formulated in the spirit of the differential equation which models Nesterov’s accelerated convex gradient method. We show that the generated trajectory converges to a critical point, if a regularization of the objective function satisfies the Kurdyka- Lojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Lojasiewicz exponent.

27. Szilárd László: Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimizationpdf
ABSTRACT:  We investigate an inertial algorithm of gradient type  in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradient method. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. Further, we provide convergence rates for the generated sequences and the function values formulated in terms of the Lojasiewicz exponent.

28. C. Alecsa, A. Viorel,  Szilárd László: A gradient type algorithm with backward inertial steps for a nonconvex minimization, (submitted 2018) pdf
ABSTRACT: We investigate an  algorithm of gradient type with a backward inertial step in connection with the minimization of a nonconvex differentiable function. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. Further, we provide convergence rates for the generated sequences and the function values formulated in terms of the Lojasiewicz exponent.

29. R.I. Boț, E.R. Csetnek,  Szilárd László:  A  primal-dual dynamical approach to  nonsmooth convex minimizations, in progress 2018, pdf
ABSTRACT:  In this paper we investigate  a  dynamical system in connection to a complexly structured minimization problem.. More precisely, the objective function of the minimization problem considered is the sum of two nonsmooth convex functions, where one of them is composed with a  continuous linear operator. We show the existence and uniqueness of strong global solutions in the framework of the Cauchy-Lipschitz Theorem and prove convergence for
the generated trajectories to the minima of the above mentioned minimization problem but also for its Fenchel dual.
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